high accuracy wadi image interpolation with local gradient features

Proceedings of 2005 International Symposium on Intelligent Signal Processing and Communication Systems

December 13-16, 2005 Hong Kong

HIGH ACCURACY WADI IMAGE INTERPOLATION WITH LOCAL GRADIENT FEATURES Shuai Yuan1, Masahide Abe1, Akira Taguchi2 and Masayuki Kawamata1 1

Department of Electronic Engineering, Graduate School of Engineering, Tohoku University 6-6-05, Aoba, Aramaki, Aoba-ku, Sendai, 980-8579 Japan 2 Department of Electrical and Electronic Engineering, Musashi Institute of Technology 1-28-1 Tamazutsumi, Setagaya-ku, Tokyo, 158--8557 Japan

sequence of low-resolution images for producing an interpolated high-resolution image. Since the interpolations based on multi-resolution pyramids or multi-image datas need heavy computations, they can not replace the bilinear or bicubic interpolations, in particular when the computational simplicity is desired. For example, in many cases the interpolator is included in end-user equipment, which should have low cost and computational simplicity. For improving the performance of the conventional linear interpolations, the conventional WaDi technique was proposed by Ramponi in [6]. Since the WaDi technique can reduce the interpolation error and has a simple computation structure, it is expected to replace the conventional linear interpolations in many applications. The conventional WaDi technique is based on the local asymmetry features of image edges. We know that the local gradient features are one kind of the important features of image edges as well as the local asymmetry features. In this paper we adopt these two kinds of features in the proposed WaDi interpolation method, and then more image edge details in the proposed WaDi interpolated images are displayed than those in the conventional WaDi interpolated images. This paper is organized as follows. The conventional WaDi interpolation is reviewed in Section 2.1. The local gradient features of digital images are described in Section 2.2. The proposed WaDi interpolation method is presented in Section 2.3. Some experimental results are given in Section 3. Finally, the conclusion is included in Section 4.

ABSTRACT In this paper, we propose a new computational method of the warped distance (WaDi) for digital image interpolation. The conventional WaDi algorithm only uses the local asymmetry features of image edges to compute the WaDi and then uses the computed WaDi in place of the interpolation operator in the linear interpolations. The local gradient features are one kind of the important features of image edges as well as the local asymmetry features. In this paper, we adopt both the local asymmetry features and the local gradient features in the WaDi computation. Experimental results show that the proposed method can obtain high accuracy interpolated images. 1. INTRODUCTION Image interpolation is a prime technique in image processing. It is used in many important applications such as digital high-definition television, big screen display, copy and print machine, medical imaging, end-user equipment and so on. The conventional image linear interpolations (e.g., the bilinear and bicubic interpolations [1]) are used widely in many applications. However, the conventional linear interpolations have a serious blurring problem, because they ignore the features of the image pixel data, such as the frequency features, the edge features, the features under multiresolution and so on. For solving the blurring problem in image interpolation, various resolution enhancement (RE) interpolation algorithms were proposed. Some of the RE interpolation algorithms [2], [3] use multiresolution pyramid representations of a lowresolution image to calculate an interpolated highresolution image. In [2], the Laplacian pyramid is used for predicting the high frequency components. In [3], a wavelet-based interpolation is presented. Another popular type of the RE interpolation algorithms [4],[5] needs a

0-7803-9266-3/05/$20.00 ©2005 IEEE.

2. WADI INTERPOLATION WITH LOCAL GRADIENT FEATURES Since any lowpass effect in the courses when we obtain natural images will change step edges into sigmoidal

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(a)

(b)

Fig. 1. Bilinear interpolation. (a) An edge model. (b) Bilinear interpolation for up-sampling.

(a)

(b)

Fig. 2. Ideal interpolation value f icideal , (a) bilinear interpolation value f i c and (b) WaDi interpolation value fˆi c .

edges, we use a sigmoidal signal as a model for image edges. The conventional linear interpolations need an upsampling distance s to estimate the unknown pixels for up-sampling. For an edge model the bilinear interpolation is shown in Fig. 1. At the position i c , the bilinear interpolation estimates the value of the interpolated pixel as

f ic

(1  s ) f i  sf i 1 ,

(1) Fig. 3. Relationship between 2-D WaDi and 1-D WaDis.

where the distance s is defined by s ic  i . However, from Fig. 1 we know that the bilinear interpolated value f ic at the position i c does not correctly represent the real ideal

value f ic

shows how the WaDi s c works. Thus, we can find s c with

. This reason makes the interpolated images

using the bilinear interpolation blurred.

fˆic

(1  sc) f i  scf i 1 f icideal .

2.1 Conventional WaDi Interpolation ideal

For obtaining the ideal interpolated value f ic

, the

(2)

The WaDi s c depends on image local features. Utilizing the local asymmetry features, the conventional WaDi interpolation defines s c as

WaDi interpolation uses the WaDi s c in place of the distance s in the conventional linear interpolations. Fig. 2

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TABLE I THE MINIMIZED MSE VALUES (128×128->256×256)

Fig. 4. Four pixel masks around the interpolated pixel.

sc

s  kAs(1  s ),

where A is the local asymmetry feature defined by

A

Bilinear

Conventional WaDi

Proposed WaDi

Airplane Cameraman Lenna Pepper Lighthouse Clock Tiffany Couple

247.8 325.1 174.1 143.2 392.0 142.6 158.4 70.4

233.8 312.7 166.5 135.6 376.1 130.3 154.1 67.2

232.4 310.4 161.9 133.9 374.6 129.2 153.4 66.9

4 with the dotted line blocks. The local gradient weights H l , H r , Vu and Vl are defined as

(3)

f i 1  f i 1  f i 2  f i , L 1

Image

Hl Hr Vu

decides the warping level. When k is large, s c may be outside of the range >0, 1@ . Since s c must be kept in >0, 1@

Vl

in real images, the outside-range s c will be clipped to 0 or 1. Two-dimensional (2-D) WaDi can be computed from one-dimensional (1-D) WaDis. Fig. 3 shows the relationship between the 2-D WaDi and the 1-D WaDis. Therefore, we have the equations for the 2-D WaDi computation as

s cy

1  ( s cy 2  s cy1 )( s cx 2  s cx1 )

1 1  D ( f i 1, j  f i  2, j  f 1i , j 1  f i  2, j 1 ) 1 1  D ( f i , j  f i , j 1  f i 1, j  f i 1, j 1 ) 1

,

1  D ( f i , j 1  f i , j  2  f i 1, j 1  f i 1, j  2 )

where the parameter

D

is in the range of >0, 1@ .

2.3 Proposed High Accuracy WaDi Interpolation From [7], we know that the local gradient weights can be used for the bilinear interpolation as

s cx1  ( s cx 2  s cx1 ) s cy1 1  ( s cy 2  s cy1 )( s cx 2  s cx1 ) s cy1  ( s cy 2  s cy1 ) s cx1

1  D ( f i , j  f i 1, j  f i , j 1  f i 1, j 1 )

(4)

where L 256 for 8-bit luminance images, and then A is limited in the range > 1, 1@ . The positive parameter k

s cx

1

(5)

f i cg, j c

.

(6)

2.2 Local Gradient Features In digital images, the local gradient features are also one kind of the important features. In [7], using the local gradient features around the interpolated pixel f ic, jc , four

Vu (1  s y ) H l (1  s x ) ˜ ˜ f i, j H l (1  s x )  H r s x Vu (1  s y )  Vl s y



Vu (1  s y ) H r sx ˜ ˜ f i 1, j H l (1  s x )  H r s x Vu (1  s y )  Vl s y



Vl s y H l (1  s x ) ˜ ˜ f i , j 1 H l (1  s x )  H r s x Vu (1  s y )  Vl s y



Vl s y H r sx ˜ ˜ f i 1, j 1 . H l (1  s x )  H r s x Vu (1  s y )  Vl s y

(7)

local gradient weights are proposed by Hwang. The four local gradient weights, H l , H r , Vu and Vl are

Using both the local asymmetry and the local gradient features, the proposed WaDi interpolation replaces s x

generated from the four pixel masks around the interpolated pixel, where the four masks are shown in Fig.

and s y in (7) with the conventional WaDis s cx and s cy , respectively. Then, we can get

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3. EXPERIMENTAL RESULTS

(a)

(b)

(c)

(d)

To do a quantitative evaluation of the proposed WaDi interpolation, we use the "imresize(I, 0.5, bilinear)" of matlab to lowpass filter the different well-known test images ( 256u 256 ) and subsample the lowpass filtered images to 1/2 times of their original size. Then, the subsampled images are interpolated to their original size using the bilinear, the proposed WaDi and the conventional WaDi interpolations with the different parameters. The minimized MSE values for each interpolation method are presented in Table I. The experimental interpolated results of "Peppers" are shown in Fig.5. From Fig.5 and Table I, we know that the proposed WaDi interpolation can get both higher interpolated accuracy and clearer edge details than those of the conventional WaDi interpolation. 4. CONCLUSION

Fig. 5. Experimental interpolated images. (a) Conventional WaDi interpolated "Peppers" image (MSE=135.6). (b) Edge components of (a) after the Laplacian filtering. (c) Proposed WaDi interpolated "Peppers" image (MSE=133.9). (d) Edge components of (c) after the Laplacian filtering.

f icp, jc

The proposed WaDi interpolation in this paper uses both the local asymmetry features and the local gradient features of an image to compute the warped distance. In this way, more image local information is considered to estimate the interpolated pixels than that in the conventional WaDi interpolation. Experiment results show that high accuracy interpolated images can be obtained using the proposed WaDi interpolation.

Vu (1  s cy ) H l (1  s cx ) ˜ ˜ f i, j H l (1  s cx )  H r s cx Vu (1  s cy )  Vl s cy



Vu (1  s cy ) H r s cx ˜ ˜ f i 1, j H l (1  s cx )  H r s cx Vu (1  s cy )  Vl s cy



Vl s cy H l (1  s cx ) ˜ ˜ f i , j 1 H l (1  s cx )  H r s c Vu (1  s cy )  Vl s cy

Vl s cy H r s cx  ˜ ˜ f i 1, j 1 . H l (1  s cx )  H r s c Vu (1  s cy )  Vl s cy

5. REFERENCES [1] W. K. Pratt, Digital Image Processing, New York: Wiley, 1991. [2] H. Greenspan, C. H. Anderson and S. Akber, “Image enhancement by nonlinear extrapolation in frequency space,” IEEE Trans. on Image Processing, vol. 9, no. 6, pp. 1035-1048, June 2000. [3] W. K. Carey, D. B. Chuang and S. S. Hemami, “Regularitypreserving image interpolation,” IEEE Trans. on Image Processing, vol. 8, no. 9, pp. 1293-1297, September 1999. [4] R. Hardie, K. Barnard and E. Armstrong, “Joint map registration and high resolution image estimation using a sequence of undersampled measured images,” IEEE Trans. on Image Processing, vol. 6, pp. 1621-1632, December 1997. [5] N. R. Shah and A. Zakhor, “Resolution enhancement of color video sequences,” IEEE Trans. on Image Processing, vol. 8, no. 6, pp. 879-885, June 1999. [6] G. Ramponi, “Warped distance for space-variant linear image interpolation,” IEEE Trans. on Image Processing, vol. 8, no. 5, pp. 629-639, May 1999. [7] J. W. Hwang and H. S. Lee, “Adaptive image interpolation based on local gradient features,” IEEE Signal Processing Letters, vol. 11, no. 3, pp. 359-362, March 2004.

(8)

Then, the proposed WaDis can be easily written as

s xP

s yP

c

c

H r s cx H l (1  s cx )  H r s cx Vl s cy , Vu (1  s cy )  Vl s cy

p

and f ic, j c can also be easily written as

f icg, jc

c c c c (1  s xP )(1  s yP ) f i , j  s xP (1  s yP ) ˜ f i 1, j

c c c c  (1  s xP ) s yP f i , j 1  s xP s yP f i 1, j 1.

(9)

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